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In this image of from BYU's x-ray diffraction
facility, x-rays arriving from the left scatter in all directions from a tiny crystal at
the center, and are then imaged by a 16-megapixel x-ray camera. The often beautiful
scattering patterns that result contain a wealth of information about the atomic structure
of the sample. The speed and sensitivity of state-of-the-art instruments like this have
revolutionized the study of crystalline materials. The stainless-steel cylinder with
lots of knobs is a double focusing Kirkpatrick-Baez mirror, which increases the
intensity of the x-ray beam by a factor of 8. The beam collimator (left),
the goniometer head (back), the microscope (45°), the
low-temperature gas nozzel (vertical), and a beam stop (right) all point towards
the location of the sample.

The spatial distribution of scattered
x-rays is closely related to the Fourier transform of the crystal's electron
density. In principle, an inverse Fourier transform can be used to
directly convert experimental scattering data into a picture of the electron
density (i.e. the atomic crystal structure). Of course, the Fourier
Transform of the electron density is a complex-valued function, only the
magnitude of which is actually measured. Hence the "phase problem" of
crystallography.

In the figure below, the lattice of
grey dots represents the Fourier transform of a simple cubic crystal with one
atom per repeating unit cell. Because the Fourier transform of any
periodic function (crystals are periodic by definition) is a discrete lattice of
uniformly spaced peaks, we call this the reciprocal lattice. The
individual peaks are called Bragg peaks. In the figure, the reciprocal
lattice and the real-space laboratory configuration have been superimposed
(don't try this at home!) to illustrate the geometry of a diffraction
experiment. Note that when the crystal rotates on the diffractometer, its
reciprocal lattice rotates with it, but around a different origin. The
well-known Bragg diffraction equation, 2dsin(q)
= l, is actually the equation for a sphere of radius
1/l with it's edge just touching the origin of
reciprocal space. We have included this imaginary "Ewald sphere" in the
figure. As the crystal rotates on the goniometer, any Bragg peak that
comes into contact with the surface of the Ewald sphere satisfies the
diffraction condition and results in a diffracted x-ray beam that leaves the
crystal in the direction of the peak. When diffracted beams intersect the
x-ray detector surface, their locations and intensities are recorded.
Click on the figure to see an interactive flash animation of a
single-crystal diffraction experiment. Choose a crystal type and
orientation to start the animation.

The animation makes the Ewald sphere
look something like a disco ball. So now we have Disco Diffraction!